1. ## Inner Product Space

Let V be an inner product space. Let $\{e_n\}_1^\infty$ be a complete orthonormal sequence in V. Suppose that $f,g \in V$ with $f = \sum_{n=1}^{\infty}c_ne_n$ and $g= \sum_{k=1}^\infty d_ke_k$. Prove that $\langle f, g \rangle = \sum_{n=1}^\infty c_n \bar{d_n}$.

My proof so far:

$\langle f, g \rangle = \langle \sum_{n=1}^{\infty}c_ne_n , \sum_{k=1}^\infty d_ke_k \rangle$

$= \sum_{n=1}^{\infty}c_n \langle e_n , \sum_{k=1}^\infty d_ke_k \rangle$

$= \sum_{n=1}^{\infty}c_n \bar{d_n} \langle e_n, e_n \rangle$

I definitely don't think I can pull that sum out in the first step because the $e_n$ has the sum with it as well. I was thinking of trying the following:

$\langle \lim_{n\to \infty} \sum_{n=1}^N c_n e_n , \lim_{k \to \infty} \sum_{k=1}^K d_k e_k \rangle$

I just don't have any idea where to go from here. I think I can pull out the limits, but I don't know how that would help.

2. Originally Posted by Benmath
Let V be an inner product space. Let $\{e_n\}_1^\infty$ be a complete orthonormal sequence in V. Suppose that $f,g \in V$ with $f = \sum_{n=1}^{\infty}c_ne_n$ and $g= \sum_{k=1}^\infty d_ke_k$. Prove that $\langle f, g \rangle = \sum_{n=1}^\infty c_n \bar{d_n}$.

My proof so far:

$\langle f, g \rangle = \langle \sum_{n=1}^{\infty}c_ne_n , \sum_{k=1}^\infty d_ke_k \rangle$

$= \sum_{n=1}^{\infty}c_n \langle e_n , \sum_{k=1}^\infty d_ke_k \rangle$

$= \sum_{n=1}^{\infty}c_n \bar{d_n} \langle e_n, e_n \rangle$

I definitely don't think I can pull that sum out in the first step because the $e_n$ has the sum with it as well. I was thinking of trying the following:

$\langle \lim_{n\to \infty} \sum_{n=1}^N c_n e_n , \lim_{k \to \infty} \sum_{k=1}^K d_k e_k \rangle$

I just don't have any idea where to go from here. I think I can pull out the limits, but I don't know how that would help.
What do you mean "pull out the limits"? Like $\displaystyle \lim_{n\to\infty}\lim_{k\to\infty}\left\langle\sum _{n=1}^{N}c_ne_n,\sum_{k=1}^{K}d_ke_k\right\rangle$?

3. I definitely think that I cannot just pull out the sum like I did in the top, so I was trying to use partial sums, but I don't know where to go from there. Yes, what I mean was pulling out the limits like you have, but I don't even know if that would be helpful.

4. There is no problem about pulling the sums out of the inner products, because the inner product is a continuous function of its two arguments. If $f = \sum_{n=1}^{\infty}c_ne_n$ then $f$ is the limit of the finite sums $\sum_{n=1}^{N}c_ne_n$. It follows that

$\langle f, g \rangle = \lim_{N\to\infty}\Bigl\langle \sum_{n=1}^{N}c_ne_n, g \Bigr\rangle = \lim_{N\to\infty}\sum_{n=1}^{N}c_n\langle e_n,g\rangle = \sum_{n=1}^{\infty}c_n\langle e_n,g\rangle,$

and similarly for the right-hand term in the inner product.